How to extrude objects during game play?
Depends on what you mean with "extrude". 

In terms of mesh-editing the answer is no. This is simply because you can't add faces to meshes from within the BGE.

As this does not help you with what you want, I suggest to get rid of the "extrude face" idea towards an "extrude effect". It does not add faces, but deforms the mesh in a way that it looks as it would do that. 

There are several ways to achieve that:


armature deformation -> it can animate scaling too
shape key deformation
scaling animation -> would effect the complete mesh
mesh manipulating via Python (not a simple task)
adding objects
...


Without more information I can't tell what option is sufficient with your situation.